Why does trying to texture paint in Blender produce no change to my model?
First, texture which is meant to be painted on is set up to not affect color rather than bump via Textures tab > Influence rollout:



Uncheck Geometry and enable Color in Diffuse section instead to paint with desired color instead of bump.

Second, brush which is used for painting on that texture uses another texture for brush pattern:



However nothing is assigned to that texture choosed to be used as a brush pattern; hence nothing will be painted.
If you'd like to paint with some brush pattern choose a desired image in Brush textures shown on the right of the screenshot above. If nothing is needed to be affecting the brush, unlink that datablock from the texture painting brush settings.